In mathematics, topological complexity of a topological space X (also denoted by TC(X)) is a topological invariant closely connected to the motion planning problem, introduced by Michael Farber in 2003.
Definition Let X be a topological space and P X = { γ : [ 0 , 1 ] → X } {\displaystyle PX=\{\gamma :[0,1]\,\to \,X\}} be the space of all continuous paths in X. Define the projection π : P X → X × X {\displaystyle \pi :PX\to \,X\times X}
by π ( γ ) = ( γ ( 0 ) , γ ( 1 ) ) {\displaystyle \pi (\gamma )=(\gamma (0),\gamma (1))} . The topological complexity is the minimal number k such that
there exists an open cover { U i } i = 1 k {\displaystyle \{U_{i}\}_{i=1}^{k}} of X × X {\displaystyle X\times X} , for each i = 1 , … , k {\displaystyle i=1,\ldots ,k} , there exists a local section s i : U i → P X . {\displaystyle s_{i}:\,U_{i}\to \,PX.}
Examples The topological complexity: TC(X) = 1 if and only if X is contractible. The topological complexity of the sphere S n {\displaystyle S^{n}} is 2 for n odd and 3 for n even. For example, in the case of the circle S 1 {\displaystyle S^{1}} , we may define a path between two points to be the geodesic between the points, if it is unique. Any pair of antipodal points can be connected by a counter-clockwise path. If F ( R m , n ) {\displaystyle F(\mathbb {R} ^{m},n)} is the configuration space of n distinct points in the Euclidean m-space, then
T C ( F ( R m , n ) ) = { 2 n − 1 f o r m o d d 2 n − 2 f o r m e v e n . {\displaystyle TC(F(\mathbb {R} ^{m},n))={\begin{cases}2n-1&\mathrm {for\,\,{\it {m}}\,\,odd} \\2n-2&\mathrm {for\,\,{\it {m}}\,\,even.} \end{cases}}}
The topological complexity of the Klein bottle is 5.
References
Farber, M. (2003). "Topological complexity of motion planning". Discrete & Computational Geometry. Vol. 29, no. 2. pp. 211–221. Armindo Costa: Topological Complexity of Configuration Spaces, Ph.D. Thesis, Durham University (2010), online
External links Topological complexity on nLab